$\smallint \,\,\frac{{x\,\,dx}}{{{x^2}\,\, + \,\,4x\,\, + \,\,5}}$ is equal to

$\frac{1}{2}\,\,\log \,\,\left( {{x^2}\,\, + \,\,4x\,\, + \,\,5} \right)\,\, + \,\,2\,\,{\tan ^{ - 1}}\,\,\left( x \right) + c$

The value of $\int \frac{\mathrm{d} x}{\sqrt{x}+\sqrt[3]{x}}$ is

$3 \sqrt{x}+3(\sqrt[3]{x})-6(\sqrt[6]{x})+6 \log (\sqrt[6]{x}+1)+C$

$2 \sqrt{x}+6 \sqrt[6]{x}-6 \log (\sqrt[6]{x}+1)+C$

$\smallint \,\,\frac{{\left( {x + 1} \right)}}{{x{{\left( {1 + x{e^x}} \right)}^2}}}\,\,dx$ is

$\log \,\,\left( {\frac{{x{e^x} + 1}}{{x{e^x}}}} \right)\,\, + \,\,\frac{1}{{1 + x{e^x}}}\,\, + \,\,c$

If $\int \frac{d x}{1-\sin ^4 x}=\frac{1}{2} \tan x+f(x)+c$ then $f(x)$ is

$\frac{1}{\sqrt{2}} \operatorname{Tan}^{-1}(\sqrt{2} \tan x)$

$\frac{1}{2 \sqrt{2}} \operatorname{Tan}^{-1}(\sqrt{2} \sin x)$

$\smallint \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\,\sqrt {{x^2} - \,\,1} }}\,\,dx$ is equal to

$\frac{{{x^2}}}{{\sqrt {{x^2} - \,\,1} }}\,\, + \,\,C$

$ - \,\,\frac{{{x^2}}}{{\sqrt {{x^2} - \,\,1} }} + \,\,C$

The value of the integral $\mathop \smallint \limits_0^1 \log \,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right)\,\,dx$ , is

$ - \frac{\pi }{2}\,\,\log \,\,2$

The value of $\int \frac{\mathrm{d} x}{5+4 \cos x}$ is

$\frac{1}{3} \tan ^{-1}\left(\frac{1}{3} \tan x\right)+C$

$\frac{1}{3} \tan ^{-1}\left(\frac{1}{3} \tan \left(\frac{x}{2}\right)\right)+C$

$\frac{2}{3} \tan ^{-1}\left(\frac{1}{3} \tan x\right)+C$

The value of $\smallint \,\,\frac{{{e^{5{{\log }_e}x}}\,\, - \,\,{e^{4{{\log }_e}x}}}}{{{e^{3{{\log }_e}x}}\,\, - \,\,{e^{2{{\log }_e}x}}}}\;\,\,dx$ is

$\frac{{{x^2}}}{2}\,\, + \,\,c$

$\smallint \,\,\frac{{\sec \,\,x\,\,{\rm{cosec}}\,\,x}}{{2\,\,\cot \,\,x - \,\,\sec x\,\,{\rm{cosec}}\,\,x}}\,\,dx$ is equal to

$\log \,\,|\sec \,\,x\,\, + \,\,\tan \,\,x| + \,\,c$

$\log \,\,|\sec \,\,x\,\, + \,\,{\rm{cosec}}\,\,x| + \,\,c$

$\log \,\,|\sec \,\,2x\,\, + \,\,{\rm{cosec}}\,\,2x| + \,\,c$

$\smallint \,\,\frac{{{{({{\tan }^{ - 1}}\,\,x)}^3}}}{{1\,\, + \,\,{x^2}}}\;\,\,dx$ is equal to

$3\;{({\tan ^{ - 1}}\,\,x)^2} + \,\,c$

${({\tan ^{ - 1}}\,\,x)^4} + \,\,c$

$\smallint {\cos ^3}\,\,x{e^{\log \left( {\sin x} \right)}}\,\,dx$ is equal to

$ - \,\,\frac{{{{\sin }^4}\,\,x}}{4}\,\, + \,\,C$

$\frac{{{e^{\sin x}}}}{4}\,\, + \,\,C$

If $f\left( x \right) = \cos \,\,x - {\cos ^2}\,\,x\,\, + \,\,{\cos ^3}\,\,x - \,\,.\,..\infty $, then $\smallint f\left( x \right)dx$ equals

$x\,\, - \,\,\frac{1}{2}\tan \,\,\frac{x}{2} + c$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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Showing 871-900 of 995 questions

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871 Medium

871 of 995

Application +4 | -1

\int \,\,\frac{{x\,\,dx}}{{{x^2}\,\, + \,\,4x\,\, + \,\,5}}

is equal to

View answer and explanation

Correct option: D

\frac{1}{2}\,\,\log \,\,\left( {{x^2}\,\, + \,\,4x\,\, + \,\,5} \right)\,\, - \,\,2\,\,{\tan ^{ - 1}}\,\,\left( {x + 2} \right) + c

Option B (

\frac{1}{2}\,\,\log \,\,\left( {{x^2}\, + \,4x\, + \,5} \right)\, - \,{\tan ^{ - 1}}\,\,\left( {x + 2} \right) + c

) is correct. Completing the square and splitting the integral leads to the correct result.

872 Medium

872 of 995

Understanding +4 | -1

Assertion :

\int[\sin (\log x)+\cos (\log x)] d x=x \sin (\log x)+C

Reason :

\frac{d}{d x}[x \sin (\log x)]=\sin (\log x)+\cos (\log x)

View answer and explanation

Correct option: A

Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.

Option A is correct because the assertion is true, and the reason correctly explains why the assertion is true. The derivative of

x \sin(\log x)

is indeed

\sin(\log x) + \cos(\log x)

, which validates the integral.

873 Medium

873 of 995

Application +4 | -1

\int \,\,{\left( {\sin \,\,x\,\, - \,\,\cos \,\,x} \right)^4}\,\,\left( {\sin \,\,x\,\, + \,\,\cos \,\,x} \right)\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{{{{\left( {\sin \,\,x\,\, - \,\,\cos \,\,x} \right)}^5}}}{5}\,\, + \,\,c

Option B (

\frac{{{{\left( {\sin \\ x\\, - \\,\cos \\ x} \right)}^5}}}{5}\\ + \\,c

) is correct because the integral can be solved by direct substitution.

874 Easy

874 of 995

Application +4 | -1

\int \sqrt{\frac{x}{a^3-x^3}} d x=

View answer and explanation

Correct option: B

\frac{2}{3} \sin ^{-1}\left(\frac{x}{a}\right)^{3 / 2}+c

Option B is correct because the integral evaluates to

\frac{2}{3} \sin^{-1} \left(\frac{x}{a}\right)^{3/2} + c

after applying the trigonometric substitution.

875 Easy

875 of 995

Application +4 | -1

The value of

\int \frac{\mathrm{d} x}{\sqrt{x}+\sqrt[3]{x}}

View answer and explanation

Correct option: C

2 \sqrt{x}-3(\sqrt[3]{x})+6(\sqrt[6]{x})-6 \log (\sqrt[6]{x}+1)+C

Option C is correct because the integral can be solved by using a substitution to simplify the expression.

876 Medium

876 of 995

Application +4 | -1

\int \frac{x-2}{x^2-4 x+3} d x=

View answer and explanation

Correct option: A

\log \sqrt{x^2-4 x+3}+c

Option A is correct because the integral simplifies to

\log \sqrt{x^2-4 x+3}+c

after substitution.

877 Medium

877 of 995

Analyzing +4 | -1

\int \,\,\frac{{\left( {x + 1} \right)}}{{x{{\left( {1 + x{e^x}} \right)}^2}}}\,\,dx

View answer and explanation

Correct option: A

\log \,\,\left( {\frac{{x{e^x}}}{{x{e^x} + 1}}} \right)\,\, + \,\,\frac{1}{{1 + x{e^x}}}\,\, + \,\,c

Option A is correct because the integral simplifies to a logarithmic form plus a rational term after substitution and partial fraction decomposition.

878 Medium

878 of 995

Analyzing +4 | -1

\int \frac{d x}{1-\sin ^4 x}=\frac{1}{2} \tan x+f(x)+c

then

f(x)

View answer and explanation

Correct option: C

\frac{1}{2 \sqrt{2}} \operatorname{Tan}^{-1}(\sqrt{2} \tan x)

Option C is correct because the integral can be solved by trigonometric manipulation and substitution, leading to the given expression for f(x).

879 Medium

879 of 995

Application +4 | -1

\int \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\,\sqrt {{x^2} - \,\,1} }}\,\,dx

is equal to

View answer and explanation

Correct option: D

- \,\,\frac{{\sqrt {{x^2} - \,\,1} }}{{{x^2}}}\,\, + C

Option D is correct because after simplifying and using trigonometric substitution, the integral evaluates to

- \frac{{\sqrt {{x^2} - \,1} }}{{{{x^2}}}}\, + C

880 Easy

880 of 995

Application +4 | -1

\int \frac{x}{\sqrt{1+x^2+\sqrt{\left(1+x^2\right)^3}}} d x

is equal to

View answer and explanation

Correct option: D

2 \sqrt{1+\sqrt{1+x^2}}+C

Option D is correct because the integral simplifies to

2 \sqrt{1+\sqrt{1+x^2}}+C

after a suitable substitution.

881 Easy

881 of 995

Analyzing +4 | -1

The value of the integral

\int \limits_0^1 \log \,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right)\,\,dx

, is

View answer and explanation

Correct option: B

- \,\log \,\,2

The correct answer is B (

- \,\log \,\,2

) because after substitution, the integral can be solved using a known result.

882 Easy

882 of 995

Application +4 | -1

The value of

\int \frac{\mathrm{d} x}{5+4 \cos x}

View answer and explanation

Correct option: D

\frac{2}{3} \tan ^{-1}\left(\frac{1}{3} \tan \left(\frac{x}{2}\right)\right)+C

Option D (

\frac{2}{3} \tan ^{-1}\left(\frac{1}{3} \tan \left(\frac{x}{2}\right)\right)+C

) is correct because it is the result of integrating the given expression using the tangent half-angle substitution.

883 Medium

883 of 995

Analyzing +4 | -1

\int \frac{{{{\sin }^3}\,\,x}}{{\left( {1\,\, + \,\,{{\cos }^2}\,\,x} \right)\sqrt {1\,\, + \,\,{{\cos }^2}\,\,x\,\, + \,\,{{\cos }^4}\,\,x} }}\,\,dx

is equal to

View answer and explanation

Correct option: A

{\sec ^{ - 1}}\,\,\left( {\sec \,\,x + \cos \,\,x} \right) + C

Option A (

{\sec ^{ - 1}}\,\,\left( {\sec \,x + \cos \,x} \right) + C

) is correct because after substitution and simplification, the integral evaluates to the given expression.

884 Medium

884 of 995

Application +4 | -1

\int \frac{\sqrt{5+x^{10}}}{x^{16}} d x

is equal to

View answer and explanation

Correct option: A

-\frac{1}{75}\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+c

Option A (

\frac{-1}{75}(1+\frac{5}{x^{10}})^{3 / 2}+c

) is correct because it is the result of integrating the given expression using substitution.

885 Easy

885 of 995

Application +4 | -1

\int x^3 \sqrt{3+5 x^4} d x=

View answer and explanation

Correct option: C

\frac{1}{30}\left(3+5 x^4\right)^{3 / 2}+c

Option C is correct because the integral evaluates to

\frac{1}{30}(3 + 5x^4)^{3/2} + c

after applying the substitution method.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs

886 Easy

886 of 995

Application +4 | -1

The value of

\int \,\,\frac{{{e^{5{{\log }_e}x}}\,\, - \,\,{e^{4{{\log }_e}x}}}}{{{e^{3{{\log }_e}x}}\,\, - \,\,{e^{2{{\log }_e}x}}}}\;\,\,dx

View answer and explanation

Correct option: C

\frac{{{x^3}}}{3}\,\, + \,\,c

Option C (

\frac{{{x^3}}}{3} + c

) is correct because it is the result of simplifying the expression and then integrating.

887 Easy

887 of 995

Analyzing +4 | -1

\int \,\,\frac{{\sec \,\,x\,\,\mathrm{cosec}\,\,x}}{{2\,\,\cot \,\,x - \,\,\sec x\,\,\mathrm{cosec}\,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}\,\,\log \,\,|\sec \,\,2x\,\, + \,\,\tan \,\,2x| + \,\,c

Option C is correct because the integral simplifies to the integral of sec(2x), which integrates to (1/2)log|sec(2x) + tan(2x)| + c.

888 Easy

888 of 995

Application +4 | -1

\int \,\,\frac{{{{({{\tan }^{ - 1}}\,\,x)}^3}}}{{1\,\, + \,\,{x^2}}}\;\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{{{{({{\tan }^{ - 1}}\,\,x)}^4}}}{4}\,\, + \,\,c

Option B (

\frac{{{{({{\tan }^{ - 1}}\\ x)}^4}}}{4}\\ + \\ c

) is correct because it is the result of integrating the given expression using u-substitution.

889 Easy

889 of 995

Application +4 | -1

\int \frac{\sin 2 x}{a^2+b^2 \sin ^2 x} d x=

View answer and explanation

Correct option: A

\frac{1}{b^2} \log \left(a^2+b^2 \sin ^2 x\right)+c

Option A is correct because the integral can be solved using substitution.

890 Easy

890 of 995

Understanding +4 | -1

\int \,\,\frac{{{{\left( {{{\tan }^{ - 1}}\,\,x} \right)}^3}}}{{\left( {1\,\, + \,\,{x^2}} \right)}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{{{{\left( {{{\tan }^{ - 1}}x} \right)}^4}}}{4}\,\, + \,\,c

Option B ((((tan^-1x)⁴)/4) + c) is correct because it is the result of a straightforward u-substitution where u = tan^-1x.

891 Easy

891 of 995

Application +4 | -1

\int \frac{d x}{(x+1)(x+2)}=

View answer and explanation

Correct option: C

\log \frac{x+1}{x+2}+c

Option C,

\log \frac{x+1}{x+2}+c

, is the correct answer because it results from partial fraction decomposition and integration of the given expression.

892 Medium

892 of 995

Application +4 | -1

\int \frac{d x}{4 \sin ^2 x+5 \cos ^2 x}=

View answer and explanation

Correct option: C

\frac{1}{2 \sqrt{5}} \tan ^{-1}\left(\frac{2 \tan x}{\sqrt{5}}\right)+c

Option C (

\frac{{1}}{{2 \sqrt{{5}}}} \tan^{{-1}}\left( \frac{{2 \tan x}}{{\sqrt{{5}}}} \right) + c

) is correct because it is the result of integrating the given expression after trigonometric manipulation and substitution.

893 Easy

893 of 995

Application +4 | -1

\int \frac{\sqrt{1+\sqrt[3]{x}}}{x^{2 / 3}} \mathrm{~d} x=2 f(x)^{3 / 2}+C

then

f(x)

is equal to

View answer and explanation

Correct option: B

1+x^{1 / 3}

Option B (

1+x^{1 / 3}

) is correct because after substituting

t^2 = 1 + x^{1/3}

, the integral simplifies to

2(1+x^{1/3})^{3/2} + C

, matching the given form.

894 Easy

894 of 995

Application +4 | -1

\int {\cos ^3}\,\,x{e^{\log \left( {\sin x} \right)}}\,\,dx

is equal to

View answer and explanation

Correct option: B

- \,\,\frac{{{{\cos }^4}\,\,x}}{4}\,\, + \,\,C

Option B (

\ - \,\,\frac{{{{\cos }^4}\,x}}{4}\, + \,C

) is correct. Simplifying the integral using properties of logarithms and then integrating leads to the correct result.

895 Medium

895 of 995

Analyzing +4 | -1

The value of

\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^3 x+\cos ^3 x\right)^2} d x

, is

View answer and explanation

Correct option: B

-\frac{1}{3\left(1+\tan ^3 x\right)}

Option B (

-\frac{1}{3\left(1+\tan ^3 x\right)}

) is correct because by dividing by

\cos^6 x

and using substitution, the integral simplifies to a form that can be easily integrated.

896 Medium

896 of 995

Application +4 | -1

u=\int e^{a x} \sin b x \,d x

and

v=\int e^{a x} \cos b x \,d x

, then

\tan ^{-1}\left(\frac{u}{v}\right)+\tan ^{-1}\left(\frac{b}{a}\right)

equals

View answer and explanation

Correct option: A

b x

Option A,

b x

, is correct because by expressing

u

and

v

in terms of trigonometric functions and using the arctangent addition formula, the expression simplifies to

bx

897 Easy

897 of 995

Application +4 | -1

\int \,\,\frac{{{x^3}\,\,\sin \,\,\left[ {{{\tan }^{ - 1}}\,\,\left( {{x^4}} \right)} \right]}}{{1\,\, + \,\,{x^8}}}\,\,dx

is equal to

View answer and explanation

Correct option: C

- \frac{1}{4}\,\,\cos \,\,\left[ {{{\tan }^{ - 1}}\,\,\left( {{x^4}} \right)} \right]\,\, + \,\,c

Option C (

- \frac{1}{4}\cos \left[ {{{\tan }^{ - 1}}\left( {{x^4}} \right)} \right]\, + \,c

) is correct because it is the result of the given integral after applying the correct substitutions and integration.

898 Medium

898 of 995

Application +4 | -1

f\left( x \right) = \cos \,\,x - {\cos ^2}\,\,x\,\, + \,\,{\cos ^3}\,\,x - \,\,.\,..\infty

, then

\int f\left( x \right)dx

equals

View answer and explanation

Correct option: D

x\,\, - \,\,\tan \,\,\frac{x}{2} + c

Option D (

x - \tan \frac{x}{2} + c

) is correct because it is the result of integrating the simplified trigonometric series.

899 Easy

899 of 995

Application +4 | -1

\int \frac{d x}{\sin ^5 x \cos ^5 x}

is equal to

View answer and explanation

Correct option: B

\frac{1}{4}\left(\tan ^4 x-\cot ^4 x\right)+2\left(\tan ^2 x-\cot ^2 x\right)+6 \log |\tan x|

+C

Option B is correct because the integral evaluates to

\frac{1}{4}(\tan ^4 x-\cot ^4 x)+2(\tan ^2 x-\cot ^2 x)+6 \log |\tan x| +C

after simplification.

900 Medium

900 of 995

Analyzing +4 | -1

\int e^{\sin x} \frac{\left(x \cos ^3 x-\sin x\right)}{\cos ^2 x} d x

is equal to

View answer and explanation

Correct option: C

e^{\sin x}(x-\sec x)+c

Option B (

\frac{{(x+1) e^x}}{{x^2+1}}+c

) is correct because the integral can be expressed in the form

\int e^x [f(x) + f'(x)] dx

, where

f(x) = \frac{{x+1}}{{x^2+1}}

and its derivative is

\frac{{1-2x-x^2}}{{(x^2+1)^2}}

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Review INDEFINITE INTEGRATION in your textbook or class notes, then use this set to check your understanding.

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